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Chapter 18: Problem 1

A \(20.0-\mathrm{L}\) tank contains 0.225 \(\mathrm{kg}\) of helium at \(18.0^{\circ} \mathrm{C}\) . The molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{inol}\) (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

Short Answer

Expert verified
There are 56.25 moles of helium in the tank. The pressure is approximately 6.82 MPa or 67.27 atm.

Step by step solution

01

Convert Mass to Moles

To convert mass to moles, use the formula: \[ n = \frac{m}{M} \]where \( m = 0.225 \text{ kg} \) and \( M = 4.00 \text{ g/mol} \) (converted to kg: \( 0.004 \text{ kg/mol} \)). \[ n = \frac{0.225}{0.004} = 56.25 \text{ moles} \]
02

Use Ideal Gas Law to Find Pressure

We use the Ideal Gas Law \( PV = nRT \) to find the pressure. Given: \( V = 20.0 \text{ L} = 0.020 \text{ m}^3 \), \( n = 56.25 \text{ moles} \), \( R = 8.314 \text{ J/(mol} \cdot \text{K)} \), \( T = 18.0 + 273.15 = 291.15 \text{ K} \). Find \( P \): \[ P = \frac{nRT}{V} = \frac{56.25 \times 8.314 \times 291.15}{0.020} \]
03

Calculate Pressure in Pascals

Perform the calculation to find \( P \) in pascals: \[ P = \frac{56.25 \times 8.314 \times 291.15}{0.020} = 6816122.4375 \, \text{Pa} \]Therefore, the pressure is approximately \( 6.82 \times 10^6 \text{ Pa} \).
04

Convert Pressure to Atmospheres

To convert pressure from pascals to atmospheres, use:\[ 1 \text{ atm} = 101325 \text{ Pa} \]\[ P_{\text{atm}} = \frac{6816122.4375}{101325} \approx 67.27 \text{ atm} \]Thus, the pressure is approximately \( 67.27 \text{ atm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moles Calculation
To determine the number of moles in a given mass of a substance, we use a simple formula from chemistry. Moles tell us how many atoms or molecules are present in a sample. The formula to calculate moles from mass is:
  • \[ n = \frac{m}{M} \]
Where:
  • \( n \) is the number of moles.

  • \( m \) is the mass of the substance in kilograms.

  • \( M \) is the molar mass of the substance in kilograms per mole.
For the specific problem involving helium gas, we start with 0.225 kg of helium. The molar mass of helium is known to be 4.00 g/mol. We need to convert this molar mass into kilograms per mole (\( 0.004 \text{ kg/mol} \) ). Now, we use the formula for calculating moles:
  • \[ n = \frac{0.225}{0.004} = 56.25 \text{ moles} \]
This means there are 56.25 moles of helium in the tank.
Pressure Conversion
Converting pressure from one unit to another can be crucial in understanding the behavior of gases. Often, pressure is measured in Pascals (Pa) or atmospheres (atm). To fully grasp pressure conversion, let's see how this is done.
Start by calculating the pressure in Pascals using the Ideal Gas Law, given by:
  • \[ PV = nRT \]
Once you've calculated the pressure (\( P \) ) from the Ideal Gas Law, you might need to convert it into a more convenient or familiar unit such as atmospheres. One atmosphere is equal to 101325 Pascals.
  • The formula for converting Pascals to atmospheres is:

  • \[ P_{\text{atm}} = \frac{P_{\text{Pa}}}{101325} \]
In this example, once we find the pressure to be approximately 6816122.4375 Pa, converting to atmospheres involves dividing by 101325:
  • \[ P_{\text{atm}} = \frac{6816122.4375}{101325} \approx 67.27 \text{ atm} \]
Pressure conversion helps make sense of high or low-pressure scenarios depending on your context or needs.
Molar Mass
Molar mass is a crucial concept in chemistry that links mass, moles, and molecules. It is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For calculating chemical quantities, it's vital to know the molar mass of the substances involved.
Understanding molar mass allows us to convert a substance's mass to moles, enabling different chemical calculations, such as those involving gas laws. To perform these conversions, one must remember:
  • The molar mass of a substance is typically found on the periodic table.

  • For our case, helium has a molar mass of 4.00 g/mol, which is equivalent to 0.004 kg/mol.
Knowing this simple conversion enables us to tackle problems involving gases like helium in terms of mass and moles.
  • For example, converting the total mass of helium in a tank to moles involves dividing by this constant molar mass.
Thus, molar mass is integral to correctly applying equations in chemistry, especially those involving the Ideal Gas Law, and grasping the basic principles of chemical conversions.

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Most popular questions from this chapter

You blow up a spherical balloon to a diameter of 50.0 \(\mathrm{cm}\) until the absolute pressure inside is 1.25 atm and the temperature is \(22.0^{\circ} \mathrm{C}\) . Assume that all the gas in \(\mathrm{N}_{2}\) is of molar mass 28.0 \(\mathrm{g} / \mathrm{mol}\) . (a) Find the mass of a single \(\mathrm{N}_{2}\) molecule. (b) How much translational kinetic energy does an average \(\mathrm{N}_{2}\) molecule have? (c) How many \(\mathrm{N}_{2}\) molecules are in this balloon? (d) What is the total translational kinetic energy of all the inolecules in the balloon?

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 \(\mathrm{K}\) . (b) Calculate the moment of inertia of an oxygen molecule \(\left(\mathrm{O}_{2}\right)\) for rotation about either the \(y\) - or \(z\) -axis shown in Fig. 18.18 . Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of \(1.21 \times 10^{-10} \mathrm{m}\) . The molar mass of oxygen atoms is \(16.0 \mathrm{g} / \mathrm{mol} .\) (c) Find the rms angular velocity of rotation of an oxygen molecule about either the \(y\) - or \(z\) -axis shown in Fig. 18.15. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery \((10,000 \mathrm{rev} / \mathrm{min}) ?\)

(a) Oxygen \(\left(\mathrm{O}_{2}\right)\) has a molar mass of 32.0 \(\mathrm{g} / \mathrm{mol}\) . What is the average translational kinetic energy of an oxygen molecule at a temperature of 300 \(\mathrm{K} ?\) (b) What is the average value of the square at a of its speed? (c) What is the root-mean-square speed? (d) What is the momentum of an oxygen molecule traveling at this speed? (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.10 \(\mathrm{m}\) on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.) (f) What is the aver- age force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 \(\mathrm{atm} ?\) (h) Compute the number of oxygen molecules that are actually contained in a vessel of this size at 300 \(\mathrm{K}\) and atmospheric pressure. (i) Your answer for part (h) should be three times as large as the answer for part (g). Where does this discrepancy arise?

We have two equal-size boxes, \(A\) and \(B\) . Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box \(A\) is at a temperature of \(50^{\circ} \mathrm{C}\) while the gas in box \(B\) is at \(10^{\circ} \mathrm{C}\) . This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? (a) The pressure in \(A\) is higher than in \(B\) . There are more molecules in \(A\) than in \(B .(\mathrm{c}) A\) and \(B\) cannot contain the same type of gas. (d) The molecules in \(A\) have more average kinetic energy per molecule than those in \(B\) . (e) The molecules in \(A\) are moving faster than those in \(B\) . Explain the reasoning behind your answers.

A metal tank with volume 3.10 L will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at a temperature of \(23.0^{\circ} \mathrm{C},\) to what temperature can the gas be warmed before the tank ruptures? You can ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.
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